sketch12 Cheerful Fact

A Cheerful Fact:
The Pythagorean Theorem
Presented By: Rachel Thysell
a2 + b2 = c2
Commonly known that a and b stand for
the lengths of the shorter sides of a right
triangle, and c is the length of the longest
side, or hypotenuse
Where did it come from?
Often associated with Pythagoras
Lived 5th Century B.C.
Founder of the Pythagorean Brotherhood
Group for learning and contemplation
 However, most commonly heard from
authors who wrote many centuries after
the time of Pythagoras
Where did it come from?
Found in ancient Mesopotamia, Egypt,
India, China, and even Greece
Known in China as “Gougo Theorem”
Oldest references are from India, in the
Sulbasutras, dating from sometime the first
millenium B.C.
The diagonal of a rectangle “produces as much
as is produced individually by the two sides.”
Famous Triples
 All the cultures
contained “triples” of
whole numbers that
work as sides
 (3,4,5) is the most
famous
 a2+b2 = 9+16 = 25 =c2
It wasn’t Pythagoras?
A common discovery
Happened during prehistoric times
Theorem came “naturally”
Independently discovered by multiple cultures
Supported by Paulus Gerdes, cultural historian
of mathematics
Carefully considered patterns and decorations used
by African artisans, and found that the theorem can
be found in a fairly natural way
Proofs of Pythagorean Theorem
Whole books devoted to ways of proving
the Pythagorean Theorem
Many proofs found by amateur
mathematicians
U.S. President James Garfield
He once said his mind was “unusually clear and
vigorous” when studying mathematics
“Square in a Square”
 Earliest proof, based
on Chinese source
 Arrange four identical
triangles around a
square whose side is
their hypotenuse
 Since all four triangles
are identical, the inner
quadrilateral is a
square
“Square in a Square”
 Big square has side a+b, so
area is equal to
(a+b)2= a2+b2+2ab
 Inner square has area c2, and
four triangles each with area of
½ab
 Big square also equals c2+2ab
 Setting them equal to each
other,
a2+b2+2ab = c2+2ab
 Therefore,
a 2 + b2 = c 2
Proof using Similar Triangles
 Most recent proof
 Triangles ACH and CBH are
similar to ABC because they
both have right angles and
share a similar angle
AC
AH
AB
AC
and
CB
HB
AB
CB
 This can be written as
AC2=ABxAH and CB2=ABxHB
 Summing these two equations,
AC2+CB2=ABxAH+ABxHB=AB
x(AH+HB)=AB2
 Therefore, AC2+BC2=AB2
Euclid’s Elements
 Most famous proof of Pythagorean Theorem
 47th Proposition states:
“in right-angled triangles the square on the side opposite
the right angle equals the sum of the squares on the
sides containing the right triangle”
 Uses areas, not lengths of the sides to prove.
Early Greek Mathematicians did not usually use
numbers to describe magnitudes
Euclid’s Proof
 The idea is to prove
that the little square
(in blue) has the
same area as the little
rectangle (also in
blue) and etc.
 He does so using
basic facts about
triangles,
parallelograms, and
angles.
Euclid continues Theorem
 There is nothing special
about “squares” in the
theorem
 It works for any geometric
figure with its base equal
to one of the sides
 Their areas equal ka2,
kb2, and kc2
 Therefore
kc2=k(a2+b2)=ka2+kb2
Distance Formula
 Also gave birth to the distance formula
 Makes classical coordinate geometry “Euclidean”
 If distance were measured some other way it would not be
Euclidean geometry
a2 + b2 = c2
Pythagorean Theorem remains one of
most important theorems
One of most useful results in elementary
geometry, both theoretically and in
practice
The End
Any Questions?
Thank You!